Nuprl Lemma : rcos-rsub
∀[x,y:ℝ]. (rcos(x - y) = ((rcos(x) * rcos(y)) + (rsin(x) * rsin(y))))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rsin: rsin(x),
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
uimplies: b supposing a,
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2
Lemmas referenced :
real_wf,
rsin_wf,
rmul_wf,
radd_wf,
rsub_wf,
rcos_wf,
req_witness,
rminus_wf,
rcos-radd,
rsin-rminus,
rcos-rminus,
rmul_functionality,
rsub_functionality,
req_weakening,
req_functionality,
real_term_value_mul_lemma,
rcos_functionality,
real_term_value_const_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
int-to-real_wf,
real_polynomial_null,
itermMultiply_wf,
req-implies-req,
req-iff-rsub-is-0,
itermMinus_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf
Rules used in proof :
because_Cache,
isect_memberEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
dependent_functionElimination,
independent_isectElimination,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
natural_numberEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. (rcos(x - y) = ((rcos(x) * rcos(y)) + (rsin(x) * rsin(y))))
Date html generated:
2017_10_04-PM-10_21_56
Last ObjectModification:
2017_08_02-AM-10_52_18
Theory : reals_2
Home
Index